Understanding Coriolis and Centrifugal Forces in Rotating Frames

Why Extra Forces Appear: Coriolis and Centrifugal Terms in a Rotating Frame

Newton’s second law is simplest in an inertial frame. Many laboratories are not inertial. Earth rotates. So do turntables, centrifuges, and weather models. Observers fixed to those frames still write a momentum equation. They must then add extra terms. Those terms are the Coriolis and centrifugal accelerations.

Start from velocity. An inertial observer and a rotating observer do not see the same time derivative of a vector. The connection is

$$\left(\frac{\mathrm{d}}{\mathrm{d}t}\right)_{\mathrm{in}} = \left(\frac{\mathrm{d}}{\mathrm{d}t}\right)_{\mathrm{rot}} + \boldsymbol{\Omega}\times$$(dtd​)in​=(dtd​)rot​+Ω×

Apply that rule twice to position. The inertial acceleration then splits. One part is the acceleration measured in the rotating frame. Two more parts involve the frame’s angular velocity $\boldsymbol{\Omega}$Ω.

The rotating-frame momentum equation for a particle of mass $m$m therefore reads

$$m\mathbf{a}_{\mathrm{rot}} = \mathbf{F} -2m\boldsymbol{\Omega}\times\mathbf{v}_{\mathrm{rot}} -m\boldsymbol{\Omega}\times(\boldsymbol{\Omega}\times\mathbf{r}) -m\dot{\boldsymbol{\Omega}}\times\mathbf{r}.$$marot​=F−2mΩ×vrot​−mΩ×(Ω×r)−mΩ˙×r.

Here $\mathbf{F}$F collects the true forces: gravity, pressure, viscosity, contact forces. The last term vanishes if $\boldsymbol{\Omega}$Ω is constant. Earth meets that approximation for most laboratory and weather problems.

The middle kinematic term is the Coriolis force. It is

$$\mathbf{F}_{\mathrm{Cor}} = -2m\boldsymbol{\Omega}\times\mathbf{v}_{\mathrm{rot}}.$$FCor​=−2mΩ×vrot​.

It appears only when the particle moves relative to the rotating frame. The force is always perpendicular to both $\boldsymbol{\Omega}$Ω and $\mathbf{v}_{\mathrm{rot}}$vrot​. It does no work. Speed can stay constant while direction changes. On Earth the horizontal Coriolis effect deflects motion to the right in the Northern Hemisphere and to the left in the Southern Hemisphere.

The remaining term is centrifugal. It is

$$\mathbf{F}_{\mathrm{cen}} = -m\boldsymbol{\Omega}\times(\boldsymbol{\Omega}\times\mathbf{r}).$$Fcen​=−mΩ×(Ω×r).

It depends on position, not on velocity. It points outward from the axis of rotation. Near Earth’s surface it slightly reduces effective gravity and tilts the local vertical. Engineers often fold it into an effective gravitational acceleration $\mathbf{g}_{\mathrm{eff}}$geff​.

Fluids use the same split per unit mass. The rotating-frame Navier–Stokes equation becomes

$$\frac{D\mathbf{u}}{Dt} = -\frac{1}{\rho}\nabla p +\nu\nabla^{2}\mathbf{u} +\mathbf{g} -2\boldsymbol{\Omega}\times\mathbf{u} -\boldsymbol{\Omega}\times(\boldsymbol{\Omega}\times\mathbf{r}).$$DtDu​=−ρ1​∇p+ν∇2u+g−2Ω×u−Ω×(Ω×r).

Weather models keep Coriolis and usually absorb centrifugal into geopotential. Large-scale winds then nearly satisfy geostrophic balance: the pressure-gradient force cancels the Coriolis force.

The terms are fictitious in one precise sense. They exist because we insisted on using a rotating observer. An inertial accountant needs only $\mathbf{F}=m\mathbf{a}$F=ma. Both descriptions agree on the path. They disagree on how to book the acceleration.

Scale analysis tells you when to keep them. The Rossby number

$$\mathrm{Ro} = \frac{U}{fL}$$Ro=fLU​

compares inertial acceleration with Coriolis acceleration. Here $f=2\Omega\sin\phi$f=2Ωsinϕ is the Coriolis parameter. Small $\mathrm{Ro}$Ro means rotation dominates. Ocean gyres and mid-latitude weather sit in that regime. A thrown baseball does not. Centrifugal effects in the lab are tiny unless $\Omega$Ω is large, as in a high-speed centrifuge.

Common mistakes mix the two terms. Centrifugal force is not “what makes a cyclone spin.” Spin-up and deflection of moving air are Coriolis effects. Centrifugal force is already present for a particle at rest in the rotating frame. Coriolis force is not.

In short, a rotating-frame momentum equation is Newton’s law plus bookkeeping. Coriolis couples to velocity and bends trajectories. Centrifugal couples to position and redefines effective gravity. Keep both when the frame’s rotation is part of the description. Drop them only after a scale estimate says they are small.